The overwhelming practical success of adaptive mesh-refinement in computational sciences
and engineering has recently obtained a mathematical foundation with a theory on optimal
convergence rates. This article first explains an abstract adaptive algorithm and its marking
strategy. Secondly, it elucidates the concept of optimality in nonlinear approximation theory
for a general audience. It thirdly outlines an abstract framework with fairly general hypotheses
(A1)-(A4) which imply such an optimality result. Various comments conclude this state of the
art overview.

All details and precise references are found in the open access article [Carsten Carstensen,
Michael Feischl, Marcus Page, and Dirk Praetorius: Axioms of adaptivity, Comput. Math. Appl.
67 (2014)] at http://dx.doi.org/10.1016/j.camwa.2013.12.003